Finite Group Databasebeta

← #36335962

Subgroups — #36335962

Order 21376.

462 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24923976C211yesmaximal, characteristic, complemented
2#24923615C211yesmaximal, characteristic, complemented
3#24923585C212yesmaximal, complemented
4#24923559C211yesmaximal, characteristic, complemented
5#24923496C212yesmaximal, complemented
6#24923466C211yesmaximal, characteristic, complemented
7#24923420C212yesmaximal, complemented
8C4×#14581938C211yesmaximal, characteristic, complemented
9D4×C167⋊C8C211yesmaximal, characteristic, complemented
10C2×#19128753C212yesmaximal, complemented
11D42×C167C211yesmaximal, characteristic
12C2×#14581938C2211yescharacteristic
13C2×#14581938C2212yescomplemented
14C22×C167⋊C8C2212yescomplemented
15#19128832C2211yescharacteristic, complemented
16#19128823—21no—
17#19128811C2212yescomplemented
18#19128809—22no—
19C4×Dic334C2211yescharacteristic, complemented
20D4×Dic167—21no—
21C2×C4⋊Dic167C2212yescomplemented
22#19128763C2212yescomplemented
23#19128763C2211yescharacteristic, complemented
24C4⋊(C167⋊C8)C2211yescharacteristic, complemented
25#19128755C2211yescharacteristic, complemented
26#19128753C2212yes—
27#19128753C2211yescharacteristic
28#19128753C2212yescomplemented
29#19128753C2212yescomplemented
30#19128753C2212yescomplemented
31C22⋊(C167⋊C8)C2212yescomplemented
32C4×C167⋊C8C2211yescharacteristic, complemented
33C22×D4×C167—21no—
34C22×D4×C167C2212yes—
35#127×C167C2211yescharacteristic
36#121×C167—21no—
37#121×C167C2212yes—
38#120×C167—22no—
39C4×D4×C167C2211yescharacteristic
40C4×D4×C167C2211yescharacteristic
41C2×Dic334—21no—
42C2×Dic334C2311yescharacteristic
43C22×Dic167—22no—
44#14581938—22no—
45#14581938—42no—
46C2×C167⋊C8—42no—
47C2×C167⋊C8C2311yescharacteristic
48C2×C167⋊C8C2312yes—
49C2×C167⋊C8C2311yescharacteristic
50C4⋊Dic167—21no—
51C4⋊Dic167—22no—
52C4⋊Dic167C2312yes—
53C4⋊Dic167C2311yescharacteristic
54Dic167⋊C4—21no—
55Dic167⋊C4—22no—
56C22⋊Dic167—22no—
57C4×Dic167—21no—
58C24×C167—22no—
59C2×D4×C167—24no—
60C2×D4×C167—22no—
61C2×D4×C167—22no—
62C2×D4×C167—21no—
63C2×D4×C167—21no—
64C2×D4×C167—42no—
65C2×D4×C167—44no—
66C2×D4×C167D411yescharacteristic
67C2×D4×C167C2312yes—
68C2×D4×C167C2311yescharacteristic
69C22×C4×C167—21no—
70C22×C4×C167C2312yes—
71C4⋊C4×C167D411yescharacteristic
72C4⋊C4×C167C2311yescharacteristic
73C22⋊C4×C167—21no—
74C22⋊C4×C167—22no—
75C22⋊C4×C167D412yes—
76C42×C167C2311yescharacteristic
77Dic334—21no—
78Dic334—21no—
79Dic334—81no—
80C2×Dic167—21no—
81C2×Dic167—21no—
82C2×Dic167—22no—
83C2×Dic167—82no—
84C167⋊C8—21no—
85C167⋊C8—21no—
86C167⋊C8—22no—
87C23×C167—21no—
88C23×C167—22no—
89C23×C167—44no—
90C23×C167—44no—
91C23×C167—82no—
92C23×C167C2×D412yes—
93D4×C167—22no—
94D4×C167—21no—
95D4×C167—22no—
96D4×C167—44no—
97D4×C167—44no—
98D4×C167—42no—
99D4×C167—81no—
100D4×C167SD1614yescomplemented
101D4×C167#4112yes—
102C2×C4×C167—22no—
103C2×C4×C167—22no—
104C2×C4×C167—21no—
105C2×C4×C167—42no—
106C2×C4×C167C2×D411yescharacteristic
107C2×C4×C167C2×D412yes—
108C2×C4×C167C2×D411yescharacteristic
109C2×C4×C167C2411yescharacteristic
110C22×C167—22no—
111C22×C167—22no—
112C22×C167—22no—
113C22×C167—41no—
114C22×C167—41no—
115C22×C167—42no—
116C22×C167—42no—
117C22×C167—42no—
118C22×C167—88no—
119C22×C167C2×SD1614yes—
120C22×C167C22×D411yescharacteristic
121C4×C167—22no—
122C4×C167—21no—
123C4×C167—41no—
124C4×C167C2×SD1612yes—
125C4×C167C2×#4111yescharacteristic
126C4×C167H(5,𝔽2)11yescharacteristic
127Dic167—41no—
128Dic167—41no—
129Dic167—42no—
130C2×C167—24no—
131C2×C167—42no—
132C2×C167—82no—
133C2×C167#54611yescharacteristic
134C2×C167C22×SD1611yescharacteristic
135C2×C167#57611yescharacteristic
136C167#329511yescharacteristic, complemented, Sylow
137#3295—1671nomaximal, Sylow
138#554—1671no—
139D42D16711yescharacteristic
140#486—1671no—
141#481—1672no—
142#476—1671no—
143#465—1672no—
144#460—1671no—
145#450—1672no—
146C4×SD16—1671no—
147C8×D4—1671no—
148C2×#102—1672no—
149C22×D4—21no—
150C22×D4C2×D16712yes—
151C2×SD16—1671no—
152C2×SD16—1672no—
153C22×C8—1672no—
154C4⋊Q8—1671no—
155#127C2×D16711yescharacteristic
156#125—3341no—
157C22⋊Q8—3342no—
158C22⋊Q8—1672no—
159#121—21no—
160#121C2×D16712yes—
161#120—22no—
162C4×Q8—1671no—
163C4×D4—3341no—
164C4×D4C2×D16711yescharacteristic
165C4×D4C2×D16711yescharacteristic
166C2×C4⋊C4—1672no—
167#106—1671no—
168#106—1672no—
169C4⋊C8—1671no—
170#103—1671no—
171#102—1671no—
172#102—1672no—
173#102—1672no—
174#102—1672no—
175#102—1672no—
176C22⋊C8—1672no—
177C4×C8—1671no—
178C24—22no—
179C2×Q8—3341no—
180C2×Q8—1671no—
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—42no—
187C2×D4—44no—
188C2×D4C22⋊D16711yescharacteristic
189C2×D4C22×D16712yes—
190C2×D4C22×D16711yescharacteristic
191C22×C4—21no—
192C22×C4—3342no—
193C22×C4C22×D16712yes—
194SD16—3342no—
195SD16—6682no—
196C2×C8—6682no—
197C2×C8—1671no—
198C2×C8—1671no—
199C2×C8—1672no—
200C4⋊C4—3342no—
201C4⋊C4—3341no—
202C4⋊C4—3341no—
203C4⋊C4—3342no—
204C4⋊C4—1671no—
205C4⋊C4—1672no—
206C4⋊C4C22⋊D16711yescharacteristic
207C4⋊C4C22×D16711yescharacteristic
208C22⋊C4—21no—
209C22⋊C4—22no—
210C22⋊C4—3342no—
211C22⋊C4C22⋊D16712yes—
212C42—3341no—
213C42C22×D16711yescharacteristic
214C23—21no—
215C23—22no—
216C23—44no—
217C23—44no—
218C23—82no—
219C23C2×C22⋊D16712yes—
220Q8—13361no—
221Q8—3341no—
222Q8—3341no—
223D4—22no—
224D4—21no—
225D4—22no—
226D4—44no—
227D4—44no—
228D4—42no—
229D4—81no—
230D4#1458193814yescomplemented
231D4#1458194912yes—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—13362no—
237C2×C4—3341no—
238C2×C4—3341no—
239C2×C4—3342no—
240C2×C4C2×C22⋊D16711yescharacteristic
241C2×C4C2×C22⋊D16712yes—
242C2×C4C2×C22⋊D16711yescharacteristic
243C2×C4C23×D16711yescharacteristic
244C8—3341no—
245C8—3341no—
246C8—3342no—
247C22—22no—
248C22—22no—
249C22—22no—
250C22—41no—
251C22—41no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—88no—
256C22C2×#1458193814yes—
257C22C22×C22⋊D16711yescharacteristic
258C4—22no—
259C4—21no—
260C4—41no—
261C4—6681no—
262C4—6681no—
263C4—6682no—
264C4C2×#1458193812yes—
265C4C2×#1458194911yescharacteristic
266C4#1912887711yescharacteristic
267C2—24no—
268C2—42no—
269C2—82no—
270C2#2492391311yescharacteristic
271C2C22×#1458193811yescharacteristic
272C2#2492407211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting, prime radicalD42×C167C2
socleC22×C167C22×D4
hypercenterD42D167
hypocenterC167#3295
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #24923976C2semidirect
2. #24923615C2semidirect
3. #24923585C2semidirect
4. #24923559C2semidirect
5. #24923496C2semidirect
6. #24923466C2semidirect
7. #24923420C2semidirect
8. C4×#14581938C2semidirect
9. D4×C167⋊C8C2semidirect
10. C2×#19128753C2semidirect
13. C2×#14581938C22semidirect
14. C22×C167⋊C8C22semidirect
15. #19128832C22semidirect
17. #19128811C22semidirect
19. C4×Dic334C22semidirect
21. C2×C4⋊Dic167C22semidirect
22. #19128763C22semidirect
23. #19128763C22semidirect
24. C4⋊(C167⋊C8)C22semidirect
25. #19128755C22semidirect
28. #19128753C22semidirect
29. #19128753C22semidirect
30. #19128753C22semidirect
31. C22⋊(C167⋊C8)C22semidirect
32. C4×C167⋊C8C22semidirect
100. D4×C167SD16semidirect
136. C167#3295semidirect
230. D4#14581938semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#249239762
#249236152
#249235852
#249235592
#249234962
#249234662
#249234202
C4×#145819382
D4×C167⋊C82
C2×#191287532
D42×C1672
#3295167